Question: There are two values of $a$  for which the equation $4x^2+ax+8x+9=0$ has only one solution for $x$. What is the sum of those values of $a$?
Solution: The quadratic formula yields \[x=\frac{-(a+8)\pm \sqrt{(a+8)^2-4\cdot 4\cdot 9}}{2\cdot 4}. \]The equation has only one solution precisely when the value of the discriminant, $(a+8)^2-144$, is 0. This implies that $a=-20$ or $a=4$, and the sum is $\boxed{-16}$.